The Compact Difference Methods For Several Classes Of Delay Partial Differential Equations

Due to the existence of the delay phenomenon, the models described by delay partial differential equations have a wide range of applications in biology, chemistry, engineering control, neural network and many other fields of sciences. It is generally difficult to obtain the analytical solutions due to the influence of the delayed terms, even if they are the simplest linear delay partial differential equations. Therefore, numerical solutions not only make the behavior of the theoretical solution intuitive and clear, but also provide important references in practice. The thesis mainly focuses on the construction of the compact difference methods and the corresponding theoretical analysis of the numerical schemes.In Chapter2, we mainly construct a class of compact difference schemes for one-and two-dimensional nonlinear hyperbolic delay partial differential equations, and study the convergency and stability of them. Furthermore, the fourth-order accuracy in both spatial and temporal directions is obtained by employing the technique of Richardson extrapolation.In Chapter3, we apply the compact finite difference scheme to the two-dimensional delay parabolic differential equations based on the Crank-Nicholson scheme, derive a type of alternate direction scheme, and analyze the convergency and stability of the scheme. Numerical experiment confirms the efficiency of the scheme.In the fourth chapter, the compact finite difference scheme is applied to one-and two-dimensional neutral delay parabolic differential equations, and the stability in L2-norm is obtained. The convergence rate are predicted by numerical studies.In the fifth chapter, we analyze a class of nonlinear delay convection-diffusion-reaction equations with constant coefficients. A special transformation is introduced to eliminate the convection term, and then, a new type of linearized compact finite difference scheme is constructed for the transformed equations, based on the second-order backward differential formula. Richardson extrapolation is applied to improve the numerical accuracy.We apply the idea in fifth chapter to the non-Fickian delay diffusion reaction equation in Chapter6, where the integral term is discreted by the composite quadrature formula. We compare our method with the other two existing algorithms in term of accuracy and CPU time. Results demonstrate that our method is efficient and practical. Some specific model problems are also simulated.We give a brief conclusion for our work and propose some future researching work in the last chapter.